Surface Area of 3D Solids

SAT Geometry and Trigonometry

In this lesson:

• Calculate surface area of prisms and cylinders
• Calculate surface area of pyramids, cones, and spheres
• Decide which surfaces count in applied problems

Your Learning Goals for This Lesson

You will be able to:

1. Find the surface area of prisms, cylinders, pyramids, cones, and spheres by summing face areas
2. Find a slant height from a vertical height; use it in a lateral area
3. Decide which surfaces count (total, lateral, one base) and solve applied problems

You Already Know What You Need

Ten-second check: the area of an 8-by-5 rectangle?

• Rectangle:
• Circle:
• Triangle:

Surface area is just adding up 2D areas — one face at a time.

A Solid Unfolds Into a Net

Surface area = total area of all faces when the solid is unfolded flat

Rectangular Prism: Three Pairs of Faces

Any right prism:

Example 1: Box with Whole Numbers

A box: , , cm. Find SA.

Top/bottom:

Front/back:

Left/right:

Example 2: Box with Decimals

A box: cm. Find SA.

Top/bottom:

Front/back:

Left/right:

Cylinder: Net is Two Circles + Rectangle

• Two circular bases:
• Lateral surface (rectangle): width , height

Cylinder Example: Total Surface Area

A cylinder has cm and cm. Find the total SA.

Answering 80π leaves out the two bases.

Lateral Only: Watch the Diameter

A cylinder has diameter 6 cm and height 10 cm. Find the lateral SA.

Using 6 as the radius gives 120π — double.

Pyramids and Cones: Slant Height

• = vertical height; = slant height, along the face
• (pyramid: is half the base edge)
• Recall: legs 3 and 4 give hypotenuse 5

Pyramid Formula: Base Plus Lateral Faces

• = area of the base
• = perimeter of the base
• = slant height (not vertical height)

For a square pyramid: and

Square Pyramid: Find the Slant Height

Base edge m, vertical height m.

Slant height: legs give

Base: Lateral:

Using as gives .

Cone: Slant Height, Then Surface Area

A cone: , vertical height . Find total SA.

Using as gives .

The Sphere: One Elegant Curved Surface

Diameter 10 in: , in²

Which Surfaces Does the Context Count?

Cylinder , . A tank open at the top has one base: .

Applied Problem: Painting a Tank

The curved side of a tank ( m, m) is painted. One litre covers m². Paint is sold in whole litres.

litres, so buy 7.

Wrapping the Box and Material Cost

The cm box from Example 1.

Wrapping, no overlap: every face, cm²

Material at $0.02 per cm²: 158 × 0.02 = $3.16

Key Takeaways and Common Mistakes

Slant height — use , not , in formulas
Diameter — halve it first:
Which surfaces — total, lateral only, or one base
Sphere vs. volume — for SA, for volume
Units — SA is always in cm², m², in²

What Comes Next: Volume of Solids

Next lesson: Volume of 3D Solids

• Same five solid types, formulas on the SAT reference sheet
• Filling containers and finding a missing dimension

Surface area = the outside. Volume = what's inside.